Reference no: EM132255410

Time Series and Prediction Computing Homework - Questions

Q1. Write a function in R called "my.garch11.filter" to compute the 1-step ahead conditional variance of a GARCH(1,1) model as well as the log-likelihood. The GARCH(1,1) conditional variance is given by

σ_t² = ω + αY_t-1² + βσ_t-1²

for t = 1 to T. To start the recursion σ₁² is set equal to the sample variance of the time series {Y_t}. The log-likelihood is defined

L_T(θ) = -(T/2)log2π - ½_t=1Σ^Tlogσ_t² - ½_t=1Σ^T Y_t²/σ_t².  

The function "my.garch11.filter"

- requires as input (in this order)

(a) vector containing the time series data Y₁, . . . , Y_T

(b) vector of GARCH(1,1) parameter (ω, α, β)' (in this order)

- returns as output a list containing

(a) sig2: vector of GARCH(1,1) 1-step conditional variances σ_t² for t = 1, . . . , T

(b) z: vector of GARCH(1,1) standardized innovations z_t = Y_t/√(σ_t²) for t = 1, . . . , T

(c) loglik: scalar log-likelihood

Q2. Write a function in R called "my.garch11.objective" to compute the GARCH(1,1) objective function to be minimized numerically by the R optimizer.

The function "my.garch11.objective"

- requires as inputs (in this order)

(a) vector containing the time series data Y₁, . . . , Y_T

(b) vector of GARCH(1,1) filter parameters (ω, α, β)' (in this order)

- returns as output a the negative of the log-likelihood

The function has to perform these checks on the input parameter

(a) check if all the parameters of the param vector are finite (check the function is.finite in R)

(b) check if the parameters of the GARCH(1,1) satisfies the following constraints: ω > 0, α > 0, β > 0, α + β < 1.

If the vector of parameters fails to satisfy any of these conditions then the function should return "infinity" (Inf).

Q3. Write a function in R called "my.garch11.mle" to estimate the GARCH(1,1) parameters using the nlminb optimizer in R.

The function "my.garch11.mle"

- requires as inputs (in this order)

(a) vector containing the time series data Y₁, . . . , Y_T

(b) a vector of initial values of the GARCH(1,1) parameter (ω, α, β)' (in this order)

- returns as output a vector containing the maximum likelihood estimate of the GARCH(1,1) parameters

The function "my.garch11.mle" has to minimise the objective function "my.garch11.objective" using the nlminb routine. When calling the nlminb routine makes sure that:

(a) You set the lower and upper bound of all the GARCH(1,1) parameters, respectively, to 10^-4 and 1.

(b) You start the minimization from the initial value provided by the function.

It is highly recommended to check the help page of the nlminb function including the examples to understand how this function precisely works.

Q4. Write a function in R called "my.garch11.stderr" to estimate the GARCH(1,1) standard errors using numerical second order derivatives.

The function "my.garch11.stderr"

- requires as inputs (in this order)

(a) vector containing the time series data Y₁, . . . , Y_T

(b) vector of containing the MLE estimates of the GARCH(1,1) parameters (ω, α, β)' (in this order)

(c) a scalar parameter h which is the step size used for the numerical approximation of the Hessian

- returns as output the vector of standard errors of the GARCH(1,1) parameters

The standard errors are computed using the asymptotic variance covariance matrix of the ML estimator based on the Hessian of the log-likelihood, that is,

(Var)^{^}(θ^{^}) = [-H_T(θ^{^})]^-1

where H_T(θ^{^}) is the Hessian evaluated at θ^{^}.

The Hessian H_T(θ^{^}) is approximated numerically using finite differences. Let ∈ be defined as θ^{^}h where θ^{^} is the ML estimator and h is the numerical step size. Let D be a 3x3 dimensional diagonal matrix whose diagonal is equal to ∈. Let e_i denote a 3 dimensional vector with 1 in position i and zero otherwise. Then, the (I, j) entry of H_t(θ^{^}) is approximated as

[HT(θ^)]i,j ≈ (L_T(θ⁽¹⁾) - L_T(θ⁽²⁾) - L_T(θ⁽³⁾) + L_T(θ⁽⁴⁾))/4∈_i∈_j

where θ⁽¹⁾, θ⁽²⁾, θ⁽³⁾, θ⁽⁴⁾ are defined as

θ⁽¹⁾ = θ^{^} + De_i + De_j

θ⁽²⁾ = θ^{^} + De_i - De_j

θ⁽³⁾ = θ^{^} - De_i + De_j

θ⁽⁴⁾ = θ^{^} - De_i - De_j

Note - The R files attached contain codes for this assignment. The code is given in attached file to test the functions. The code is also used to evaluate the accuracy of the submission.

Attachment:- Assignment Files.rar